Section 1.5: Relative Velocity

Relative velocity is the velocity of an object as observed from a different frame of reference. If two objects move in a line or plane, understanding relative motion helps predict positions and velocities from different perspectives.

Key Concept:
  • Velocity of object A relative to object B: \( \vec{v}_{A/B} = \vec{v}_A - \vec{v}_B \)
  • Velocity of object B relative to object A: \( \vec{v}_{B/A} = \vec{v}_B - \vec{v}_A = -\vec{v}_{A/B} \)
  • For perpendicular motion, use vector addition: \( v_{rel} = \sqrt{(v_{Ax}-v_{Bx})^2 + (v_{Ay}-v_{By})^2} \)

Example 1

A boat moves at 5 m/s relative to water, and the river flows at 3 m/s. Find the boat's velocity relative to the shore if it moves upstream and downstream.

Upstream: \( v_{boat/shore} = 5 - 3 = 2 \text{ m/s} \)

Downstream: \( v_{boat/shore} = 5 + 3 = 8 \text{ m/s} \)

Example 2

A plane flies at 200 m/s relative to air. Wind blows perpendicular to the plane’s path at 50 m/s. Find the resultant velocity of the plane relative to the ground.

\( v_{resultant} = \sqrt{200^2 + 50^2} = \sqrt{42500} \approx 206.2 \text{ m/s} \)

Practice Problems

  1. A swimmer swims across a 50 m wide river at 2 m/s, while the river flows at 1 m/s. Find the resultant speed and direction.
  2. A train moves at 60 m/s relative to the tracks. A person walks in the train at 2 m/s in the same direction. Find the person’s velocity relative to the ground.
  3. A boat moves at 10 m/s in still water. The river flows at 3 m/s perpendicular to boat's motion. Find resultant velocity and angle.
  4. A plane flies 300 km/h east. Wind blows 40 km/h north. Find the resultant speed and direction.
  5. A person runs at 5 m/s on a train moving at 20 m/s in same direction. Determine relative velocity to a stationary observer.
  6. A car travels at 25 m/s, and another car moves at 15 m/s in opposite direction. Find relative speed.
  7. A boat moves upstream with 6 m/s relative to water. If river flows at 2 m/s, find time to cross 120 m wide river (perpendicular to flow).
  8. A plane heads north at 180 km/h while wind blows east at 60 km/h. Find resultant velocity.
  9. A bird flies 12 m/s north; wind is 5 m/s east. Determine the bird's resultant velocity and angle to north.
  10. Two cars move toward each other at 20 m/s and 15 m/s. Find their relative velocity.
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